Nearly reducible finite Markov chains: Theory and algorithms

Abstract

This thesis is the result of my own work and includes nothing which is the outcome of work done in collaboration except as declared in the Preface and specified in the text. I further state that no substantial part of my thesis has already been submitted, or, is being concurrently submitted for any such degree, diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the Preface and specified in the text. It does not exceed the prescribed word limit for the relevant Degree Committee.

Kinetic Monte Carlo simulations for heterogeneous catalysis: Fundamentals, current status, and challenges

“…An early such scheme is the so-called absorbing Markov chain KMC (AMC-KMC) framework. [163][164][165] Fundamental to this scheme is the concept of a superbasin, i.e. a set of basins (KMC states) connected by fast (low energy-barrier) quasi-equilibrated elementary events.…”

Selecting Features for Markov Modeling: A Case Study on HP35

“…We have employed a correlation analysis strategy to identify appropriate features and discussed in detail two main types, that is, interresidue contacts and backbone dihedral angles. A successful state partitioning should provide conformational states that are clearly discriminated by some descriptors derived from the features, such as the mean and the variance of the contact cluster distributions (Figure ). The quality of the state partitioning can be furthermore assessed by the MPP dendrogram (Figure ), which reveals potential conformational heterogeneity of the microstates being merged to a macrostate. Dynamical corrections such as low-pass filtering of the feature trajectory and the optimal projection of microstate dynamics onto macrostate dynamics − may significantly improve the Markovianity of the MSM. …”

“…Dynamical corrections such as low-pass filtering of the feature trajectory and the optimal projection of microstate dynamics onto macrostate dynamics − may significantly improve the Markovianity of the MSM.…”

“…However, this assumption rests on a time scale separation between intrastate and interstate dynamics, which is often only approximately true for macrostates constructed from MPP. As a remedy, we may calculate the macrostate transition matrix by invoking an approach to optimally project the microstate dynamics onto the macrostate dynamics, − as achieved by the method of Hummer and Szabo . Since the direct calculation yields on average 40% shorter time scales, while the latter virtually reproduces the time scales of the microstates (Figure a), we generally employ the projection method in the following.…”

The Energy Landscape Perspective: Encoding Structure and Function for Biomolecules

“…For example, we have applied kinetic path sampling to analyse the FPT for tryptophan zipper peptide tryptophan zipper1 (TZ1) (Cochran et al, 2001), revealing a bimodal distribution, which can be directly interpreted in terms of the underlying energy landscape . Additional properties such as committor probabilities, reactive visitation probabilities, and productive paths can also be obtained (Sharpe and Wales, 2021a;Sharpe and Wales, 2021b), providing a clear resolution of dynamical bottlenecks.…”